FOM: What we mean by "nonstandard analysis"

[David Ross]

I think we are conflating different ideas of "nonstandard analysis"
here: (1) Nonarchimedean extensions (or variants) of R (e.g., the
posts about formal series models, and Conway's surreal numbers);
(2) NSA as an artifact of mathematical logic;
(3) NSA as a tool in general mathematics.

(1) has already been addressed by others.  W/r to (2) and (3), it is
worth noting that questions about whether adding an ST predicate is
conservative, or the reverse mathematics of formal NSA, etc., are
quite interesting, but pretty irrelevant to the modern practice of
NSA. Most of the interesting applications of NSA in the last 20 years
or so have involved 'nonstandard hull' constructions instead of
transfer alone (Joram Hirschfeld's proof being in my opinion being one
of three major exceptions), and classical syntactic questions don't
seem to be very relevant for these applications.

Now, I don't know how important my point about (3) really is as
applied to FOM as it seems to be construed on this list; I bring it up
mainly to reinforce what Barwise has already alluded to, namely that
big nonstandard models are important for modern practice (especially
in probability and functional analysis).  (BTW, anyone who wants to
know what I mean by 'modern practice' might take a look at the recent
NATO volume edited by Arkeryd, Cutland, and Henson [Nonstandard
analysis: theory and applications].)  Of course, there *are* questions
arising from this practice which would generally (perhaps not on this
list) be considered 'foundational'; for example, a problem arising in
stochastic geometry led me to formulate a question about nonstandard
measures which was finally answered by Jin and Shelah; they showed
that the answer depends on the underlying set theory, for example if
CH fails then the answer was undecidable in ZFC.

- David Ross (ross at tarski.math.hawaii.edu)